The claimed advancements described herein relate to a system and associated methodology of detecting curvilinear objects in an image. More specifically, an apparatus and associated methodology are provided for performing a feature adapted Beamlet transform for the detection of curvilinear objects in a noisy image via a software and/or hardware implementation.
In image processing systems computer vision applications and the like, the detection of curvilinear objects is often times desired. Such objects occur in every natural or synthetic image as contours of objects, roads in aerial linear imaging or DNA filaments in microscopy applications. Currently, there is no known methodology in which a steerable filter may be leveraged to employ line segment processing methodologies such as beamlet methods for representing curvilinear objects carrying a specific line-profile.
Curvilinear objects are considered as 1 dimensional manifolds that have a specific profile running along a smooth curve. The shape of this profile may be an edge or a ridge-like feature. It can also be represented by more complex designed features. For example, in the context of DNA filament analysis in fluorescent microscopy, it is acceptable to consider the transverse dimension of a filament to be small relative to the PSF (point spread function) width of the microscope. Hence, the shape of the profile may be accurately approximate by a PSF model.
One way to detect curvilinear objects is to track locally the feature of the curve-profile; linear filtering or template matched filtering are well-known techniques for doing so. Classical Canny edge detector and more recently designed detectors are based on such linear filtering techniques. They involve the computation of inner-products with shifted and/or rotated versions of the feature template at every point in the image. High response at a given position in the image means that the considered area has a similarity with the feature template. Filtering is usually followed by a non-maxima suppression and a thresholding step in order to extract the objects. The major drawbacks of such approaches come from the fact that linear filtering is based on local operators. Hence it is highly sensitive to noise but not sensitive to the underlying smoothness of the curve, which is a typical non-local property of curvilinear objects.
Alternatively, the Radon transform is a powerful non-local technique which may be used for line detection. Also known as the Hough transform in the case of discrete binary images, it performs a mapping from the image space into a line parameter space by computing line integrals. Formally, given an image f defined on a sub-space of R2, for every line parameter (ρ, θ), it computes
                              φ          ⁡                      (                          ρ              ,              ϑ                        )                          ⁢                              ∫                          -              ∞                        ∞                    ⁢                                    ∫                              -                ∞                            ∞                        ⁢                                          f                ⁡                                  (                                      x                    ,                    y                                    )                                            ⁢                              δ                ⁡                                  (                                      ρ                    -                                          x                      ⁢                                                                                          ⁢                                              cos                        ⁡                                                  (                          θ                          )                                                                                      -                                          y                      ⁢                                                                                          ⁢                                              sin                        ⁡                                                  (                          θ                          )                                                                                                      )                                            ⁢                                                          ⁢                              ⅆ                x                            ⁢                                                          ⁢                                                ⅆ                  y                                .                                                                        (        1        )            Peaks in the parameter space reveals potential lines of interest. This is a very reliable method for detecting lines in noisy images. However, there are several limitations. First, direct extension of that method to detect more complex curves is unfeasible in practice for it increases the complexity exponentially by adding one dimension to the parameter space. In addition, Radon transform computes line integrals on lines that pass through the whole image domain and does not provide information on small line segments.
Given an image of N×N pixels, the number of possible line segments defined is in O(N4). Direct evaluation of line integrals upon the whole set of segments is practically infeasible due to the computational burden. One of the methodologies proposed to address this problem is the Beamlet transform. It defines a set of dyadically organized line segments occupying a range of dyadic locations and scales, and spanning a full range of orientations. This system of line segments, called beamlets, have both their end-points lying on dyadic squares that are obtained by recursive partitioning of the image domain. The collection of beamlets has a O(N2 log(N)) cardinality. The underlying idea of the Beamlet transform is to compute line integrals only on this smaller set, which is an efficient substitute of the entire set of segments for it can approximate any segment by a finite chain of beamlets. Beamlet chaining technique also provides an easy way to approximate piecewise constant curves.
Formally, given a beamlet b=(x, y, l, θ) centered at position (x,y), with a length l and an orientation θ, the coefficient of b computed by the Beamlet transform is given by
                              Φ          ⁡                      (                          f              ,              b                        )                          =                              ∫                                          -                1                            /              2                                      1              /              2                                ⁢                                    f              ⁡                              (                                                      x                    +                                          γcos                      ⁡                                              (                        θ                        )                                                                              ,                                      y                    +                                          γsin                      ⁡                                              (                        θ                        )                                                                                            )                                      ⁢                                                  ⁢                                          ⅆ                γ                            .                                                          (        2        )            
Equation (2) is closely related to equation (1) since Beamlet transform can be viewed as a multiscale Radon transform; they both integrate image intensity along line segments. However, they do not take into account any line-profile. It implies that the Radon and Beamlet transforms are not well-adapted to represent curvilinear objects carrying a specific line-profile.
Accordingly, a feature-adapted Beamlet transform is provided to represent curvilinear objects of a specific line profile.